A rescue RNA is an mRNA that does not have the target of the knockdown agent (e.g. siRNA, LNA, Morpholino, etc.) but codes for the protein for which synthesis is suppressed by the knockdown agent.

I'll give an example for Morpholino oligos, as this is the technology I'm most familiar with. Say you target an mRNA for the oep gene using a Morpholino that binds in the mRNAs 5'-UTR to block the initiation complex so that it cannot reach the start AUG and begin translation. You inject the oligo into a frog embryo at the single-celled stage and it changes the frog's phenotype as it develops. You are not sure whether the change in the phenotype is due to successful knockdown of the target gene or an unexpected interaction with an off-target RNA. To check this, you construct an RNA that has the coding sequence of the oep gene but has a 5'-UTR from another gene. Now this engineered RNA lacks the target for the Morpholino but still can produce oep on a ribosome. By coinjecting the engineered RNA with the oep Morpholino, you test specificity of the original knockdown. If the coinjected embryo develops normally, without the oep phenotype, then the phenotype you observed in the Morpholino-only experiment was caused by knockdown of ope; if the (putative) oep phenotype still occurs in the coinjected embryo, then either the phenotype is not due to knockdown of oep (likely instead an off-target RNA interaction) or some aspect of the rescue experiment was messed up (for instance, the rescue RNA degraded before injection). The main technical hurdle for this experiment is to deliver the rescue RNA into all of the cells that undergo the knockdown.

So, the rescue experiment is a specificity control experiment used to show that an observed change in cells/embryos/etc. is in fact due to the suppression of synthesis of a particular protein.